"The Apocalypse, or Revelation to John, the last book of the Bible, is one of the most difficult to understand because it abounds in unfamiliar and extravagant symbolism, which at best appears unusual to the modern reader."

The 4 living creatures and 24 elders are mentioned numerous times in Revelation.

In Revelation 4:

4 Surrounding the throne I saw twenty-four other thrones on which twenty-four elders sat, dressed in white garments and with gold crowns on their heads. 5
From the throne came flashes of lightning, rumblings, and peals of thunder. Seven flaming torches burned in front of the throne, which are the seven spirits of God. 6 In front of the throne was something that resembled a sea of glass like crystal.
In the center and around the throne, there were four living creatures covered with eyes in front and in back. 7 The first creature resembled a lion, the second was like a calf, the third had a face like that of a human being, and the fourth looked like an eagle in flight. 8 The four living creatures, each of them with six wings, were covered with eyes inside and out. Day and night they do not stop exclaiming:
“Holy, holy, holy is the Lord God almighty,
who was, and who is, and who is to come.”

9 Whenever the living creatures give glory and honor and thanks to the one who sits on the throne, who lives forever and ever, 10 the twenty-four elders fall down before the one who sits on the throne and worship him, who lives forever and ever. They throw down their crowns before the throne, exclaiming:
11 “Worthy are you, Lord our God,
to receive glory and honor and power,
for you created all things;
because of your will they came to be and were created.”

In Revelation 5:

6 Then I saw standing in the midst of the throne and the four living creatures and the elders, a Lamb that seemed to have been slain. He had seven horns and seven eyes; these are the [seven] spirits of God sent out into the whole world. 7 He came and received the scroll from the right hand of the one who sat on the throne. 8 When he took it, the four living creatures and the twenty-four elders fell down before the Lamb. Each of the elders held a harp and gold bowls filled with incense, which are the prayers of the holy ones.

The creatures and elders are directly mentioned again in 5:11 and 5:14.

In Revelation 6:

1 Then I watched while the Lamb broke open the first of the seven seals, and I heard one of the four living creatures cry out in a voice like thunder, “Come forward.” 2 I looked, and there was a white horse, and its rider had a bow. He was given a crown, and he rode forth victorious to further his victories.

This pattern is repeated thrice more in the next passages, 6:3-8. The Lamb breaks the next seal, the next living creature cries out "Come forward", and the next horse and rider emerge.

Then in Revelation 7, 14, 15, 19:

7:11 All the angels stood around the throne and around the elders and the four living creatures. They prostrated themselves before the throne, worshiped God,

14:3 They were singing [what seemed to be] a new hymn before the throne, before the four living creatures and the elders. No one could learn this hymn except the hundred and forty-four thousand who had been ransomed from the earth.

15:7 One of the four living creatures gave the seven angels seven gold bowls filled with the fury of God, who lives forever and ever.

19:4 The twenty-four elders and the four living creatures fell down and worshiped God who sat on the throne, saying, “Amen. Alleluia.”

What does Christian scripture, major/longstanding tradition, or Church teaching tell us about the four living creatures and the twenty-four elders?

Note that I am not asking for personal opinions or interpretations of these passages. Please do not post answers that "reason through" the passages, saying "because of X, the elders are probably Y", unless you are quoting or carefully explaining some source that is considered reasonably popular and scholarly by some body of Christians.

5 Answers
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For the person asking about the four living creatures,think on this at that time John is seeing things from maybe at least one billions years ago to his days and then to the future,after the future he is also seeing what it will be like after we get to Heaven.
So to answer your question me been a 65 year old man and seeing all the advances we have made
what John saw were 4 creatures each representing the master of their species and very possible
what we consider a computer remember John saw many eyes very well could be all the flashing lights,this my Brother comes as revelation.

The Four living creatures are Four Angels mention In the Book Of Enoch...Chapter 40.....[Chapter 40]

1 And after that I saw thousands of thousands and ten thousand times ten thousand, I saw a multitude 2 beyond number and reckoning, who stood before the Lord of Spirits. And on the four sides of the Lord of Spirits I saw four presences, different from those that sleep not, and I learnt their names: for the angel that went with me made known to me their names, and showed me all the hidden things. 3 And I heard the voices of those four presences as they uttered praises before the Lord of glory. 4,5 The first voice blesses the Lord of Spirits for ever and ever. And the second voice I heard blessing 6 the Elect One and the elect ones who hang upon the Lord of Spirits. And the third voice I heard pray and intercede for those who dwell on the earth and supplicate in the name of the Lord of Spirits. 7 And I heard the fourth voice fending off the Satans and forbidding them to come before the Lord 8 of Spirits to accuse them who dwell on the earth. After that I asked the angel of peace who went with me, who showed me everything that is hidden: 'Who are these four presences which I have 9 seen and whose words I have heard and written down?' And he said to me: 'This first is Michael, the merciful and long-suffering: and the second, who is set over all the diseases and all the wounds of the children of men, is Raphael: and the third, who is set over all the powers, is Gabriel: and the fourth, who is set over the repentance unto hope of those who inherit eternal life, is named Phanuel.' 10 And these are the four angels of the Lord of Spirits and the four voices I heard in those days.

The four living creatures can be identified as the last priest king given understanding of the most high, and only Lord God's will. Lion- king, Ox- priestly sacrifice, Eagle as it is soaring- one of understanding (includes the many eyes within).
They four are the one mightiest angel (Chayot), who authored the small scroll; sweet in the mouth, but bitter in the belly. He would absolutely have to be foretold of throughout the entire old and new testaments. Such mystery as was commanded to be with-held. Yet, in the last day, a more pure and sure definition of the Fathers holiest name, and will, would be disclosed.

It seems that it would not make much sense for the four living creatures to represent the four evangelists, considering John was one of the four, and John was the one looking into heaven to witness these four living creatures. Why would John be represented in heaven during his own vision? I think that places the importance of John on a higher level than it should.

Christian tradition going back as far as St Irenaeus has interpreted these four creatures as standing for the four evangelists because they "carry" Jesus Christ to men. The one with the face of a man is St Matthew, who starts his book with the human genealogy of Christ; the lion stands for St Mark: his Gospel begins with the voice crying in the wilderness (which is where the lion's roar can be heard); the ox is a reference to the sacrifices in the temple of Jerusalem, which is where St Luke begins his account of Christ's life, and the eagle represents St John, who soars to the heights to contemplate the divinity of the Word.

Saint Irenaeus lived 130-202. However, the notes on Revelation 4 in NABRE seem to clarify this, and they read:

Universal creation is represented by the four living creatures (Rev 4:6–7) [...]

Lion…calf…human being…eagle: these symbolize, respectively, what is noblest, strongest, wisest, and swiftest in creation. Calf: traditionally translated “ox,” the Greek word refers to a heifer or young bull. Since the second century, these four creatures have been used as symbols of the evangelists Mark, Luke, Matthew, and John, respectively.

Twenty-four elders: these represent the twelve tribes of Israel and the twelve apostles; cf. Rev 21:12–14.

Revelation 21:12-14 reads:

12 It had a massive, high wall, with twelve gates where twelve angels were stationed and on which names were inscribed, [the names] of the twelve tribes of the Israelites.
13 There were three gates facing east, three north, three south, and three west.
14 The wall of the city had twelve courses of stones as its foundation, on which were inscribed the twelve names of the twelve apostles of the Lamb.